Extended finite elements for problems with Voids
Tipus de documentProjecte Final de Màster Oficial
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The Finite Element Method (FEM) is currently widely used for the numerical so- lution of boundary value problems de ned by Partial Di erential Equations. It has been successfully applied in many areas of applied sciences and engineering. Nev- ertheless, standard FEM are not well suited for some applications such as problems with moving boundaries or interfaces. In those problems standard FEM requires the mesh to be adapted to the interface or moving boundary, requiring continuous remeshing as time evolves, leading to high computational cost and lost of accuracy due to the projection of the solution from one mesh to another. eXtended Finite Elements (X-FEM) overcome this limitation. X-FEM is able to handle interfaces and boundaries inside the nite elements, getting rid of the adap- tation of the mesh. The computational mesh covers the domain and (1) the solution is enriched to describe discontinuities inside the elements, (2) the numerical inte- gration in each element is adapted to integrate only inside the domain. X-FEM is currently a widely used technique for the solution of problems with cracks, two- phase ow problems or problems with voids. This work focuses on problems with voids. In this case no re nement is needed and attention focuses on adapting the numerical integration. A computation mesh including the domain and not adapted to the voids boundary is considered. A level set function is used to de ne the interface corresponding to the voids boundaries. Using the level set value at the nodes, elements are classi ed as: interior, in void or cut by the interface (the void boundary). For interior elements standard FEM integration is used, elements inside the void are not considered for integration and a special numerical quadrature is de ned for elements cut by the interface. Each element cut by the interface is split in subtriangles and a numerical quadrature is considered for the subtriangles in the domain. In this work an X-FEM code for voids has been developed from a standard FEM code. The developed routines aim to be a library for the X-FEM solution of two- phase ow problems.